SOVIET PHYSICS JE TP
VOLUME 30, NUMBER 4
APRIL, 1970
QUASILINEAR RELAXATION OF AN ELECTRON BEAM IN AN INHOMOGENEOUS
BOUNDED PLASMA
v
B. N. BREIZMAN and D. D. RUYTOV
Nuclear Physics Institute, Siberian Branch, U.S.S.R. Academy of Sciences
Submitted April 28, 1969
Zh. Eksp. Teor. Fiz. 57, 1401-1410 (October, 1969)
The influence of plasma inhomogeneities on the solution of the stationary boundary problem concerning
the interaction between an electron beam and a plasma is investigated. It is shown that the inhomogeneity leads to two important effects. Firstly, if the plasma contains regions in which the density
increases in the direction of motion of the beam, then quasilinear relaxation causes the appearance
of accelerated electrons. Secondly, the quasilinear relaxation in an inhomogeneous plasma proceeds
at a much slower rate than in a homogeneous one (in the absence of "trapped" oscillations).
It is clear that for sufficiently small values of the
density of the beam the spread in the velocities liv when
the beam leaves the plasma is small in comparison with
vo, so that the characteristic increment in the beam instability y can be estimated by means of the formula
INTRODUCTION
FOR a theoretical description of experiments on the
stationary injection of weak electron beams into a
plasma it is necessary to solve a boundary quasilinear
problem. In the homogeneous plasma approximation
such a problem was investigated in detail in the papers
by Fat'nberg and Shapiro[ll and Tsytovich[2 J. On the
other hand, in[3J it was established that in a quasilinear
problem with initial conditions if even very insignificant
inhomogeneities of the density are taken into account
this leads to a cardinal change in the nature of the quasilinear relaxation. Therefore, the problem arises of the
influence of plasma inhomogeneities on the solution of
the boundary problem. In the present communication we
show that under definite conditions this influence turns
out to be very significant.
For the sake of simplicity we assume that the parameters of the plasma and of the beam depend only on the
one coordinate x directed along a strong constant magnetic field, and that the characteristic wavelength of the
Langmuir oscillations induced by the beam is small
compared to the scale of the plasma inhomogeneities
and to the size of the region of quasilinear relaxation.
Under these conditions it is possible, firstly, to restrict oneself to the one-dimensional variant of the
quasilinear theory and, secondly, to utilize the quasiclassical approximation for the Langmuir oscillations.
With respect to the properties of the beam we assume that its density nb is small compared to the plasma
density n, while its initial velocity vo considerably exceeds the thermal velocity of the plasma electrons vT.
We neglect the initial spread of the velocities in the
beam.
Before we proceed to solve the problem formulated
above we obtain some simple estimates which refer to
the relaxation of an electron beam in a homogeneous
plasma. In particular, we find the spread of the velocities in the beam when it leaves the plasma as a function
of the density of the beam. The corresponding results
will contain nothing that is in principle new compared
with[ 1 J, but the qualitative derivation utilized by us will
enable us to understand in a simpler manner the special
features of relaxation in an inhomogeneous plasma.
nb (
v~-n
Vo )
!W
2
oop,
where wp = (41Tne 2 /m) 112 is the electronic plasma frequency.
The oscillations to which the beam gives rise propagate into the interior of the plasma with the group velocity v g ~ vT/vo. If we denote by L the length of the region occupied by the plasma, then the time r during which
the oscillations are amplified by the beam 11 can be estimated as L/v . During the time r the energy of the oscillations mu~ grow from the initial (thermal) level to
a level which significantly exceeds the initial one (otherwise the oscillations could not react back on the beam).
This condition can be written in the following manner:
(1)
where A is the logarithm of the ratio of the final energy
density of the oscillations to the initial one. Since this
ratio is very large, then A depends weakly on the final
level of oscillations, and usually one can assume that
A~ 10-20.
The spread of the velocities when the beam leaves the
plasma is established just at such a level at which relation (1) is satisfied:
yt ~ nb
n
oopL vo2 vo2 ~ A,
vo v:r2 Av2
whence we have
(2)
Here we have denoted by I and T respectively the energy of the electrons in the beam and the temperature
1>We make the natural assumption that at the boundary of the
plasma the oscillations are absorbed and, therefore, we do not take into
account the possibility of repeated traversal by the oscillation of the
plasma interval. In the case when the boundaries of the plasma reflect
oscillations perfectly, and the plasma density is homogeneous the problem concerning the relaxation was solved in [4 ).
759
D. N. BREIZMAN and D. D. RUYTOV
760
n(z/
of the electronics of the plasma. Of course the estimate
so obtained is valid only under the condition t:.v/vo « 1.
The relaxation process in the case under consideration has one characteristic property: it does not lead to
the appearance of electrons with velocities v > vo.
FIG. I. Density profiles considered in
Therefore, the estimate (2) obtained above for the spread this paper.
of the velocities in the beam must be interpreted in the
sense that the distribution function for the electron beam
as it leaves the plasma differs from zero over a range
of velocities Vo- t:.v < v < Vo·
From formula (2) it may be seen that as the density
of the beam nb is increased the spread of the velocities
t:.v increases and for
(3}
becomes of order of Vo, i.e., the beam has time over the
length L to relax to a plateau state:
I=
{ 2nb/Vo,
0,
v
v
< Vo
>
Vo
(4)
where f is the distribution function for the electrons of
the beam. But if the concentration of the beam becomes
greater than the critical value nb 0 which is determined
by relation (3}, then the beam has time to go over into
the plateau state (4} over a certain distance l < L, after
which the relaxation comes to an end2 >. In order to find
l one should in formula (2) replace L by l, and set t:.v/vo
equal to unity. As a result of this we find that
(5}
We now proceed to investigate the relaxation of the
beam in an inhomogeneous plasma. We consider three
density profiles typical for experiments (cf., Fig. 1, a-c)
where for concreteness we assume that the characteristic scale of the inhomogeneity is in order of magnitude
equal to L-the size of the plasma interval.
In the case of a homogeneous plasma the stationary
nature of the solution of the boundary problem is
guaranteed by the fact that the generation of oscillations in the case of beam instability is compensated by
their being carried in the direction of the beam[ 11 • A
stationary solution is also possible in the cases shown
in Fig. 1 a-c since in the case of such density profiles
the oscillations are not "trapped" within the volume of
the plasma. But if the function n(x) has sufficiently deep
minima (Fig. 1 d), then the oscillations are "trapped"
in the corresponding potential wells and the instability
is not compensated by the displacement of the oscillations3>. In such a case the boundary problem has no
stationary quasilinear solution.
If the characteristic dimension and the characteristic
amplitude of the small-scale inhomogeneities in the
density are respectively equal to a and t:.n, then the condition for the absence of minima superimposed on a "
background of smoothly varying functions n(x} shown in
Fig. 1 a-c will be given by t:.n/n « a/L. In what follows
we shall consider that this condition is satisfied.
for nb > nb 0 the quantity t:J.v can no longer be determined by means of formula (2).
3>In particular this refers to those potential wells which are situated
close to the point x =0, where the beam has not yet been spread out in
energy and the increment is maximal.
2>Naturally,
t z
1. THE PLASMA DENSITY DECREASES MONOTONIC-
ALLY IN THE DIRECTION OF INJECTION OF THE
BEAM
In this case as the oscillations generated by the beam
are propagated in the direction x > 0 they enter a region of lower values of the density, and as a result of this
their phase velocity diminishes. We find the change in
the phase velocity of the oscillations Vph over a distance
t:.x. In order to do this we make use of the dispersion
relation for the Langmuir oscillations which has the
form
co= illp(x) [1
+ 12(vT! Vph) 2].
8
Since the frequency of a wave propagating in a medium
with stationary parameters does not vary we can write
VT1 !:J.Vph
3 V~ ) drop
Aoo= ( 1+--- --t:J.x-Soop----=0.
2 Vph 2 dx
Vph 2 Vp~
From this we can first of all see that t:J.vph < 0 for
dwp/dx < 0. Further, since Vph- Vo
VT, ldwp/dxl
- wp/L, then we have
»
lix vo2
l!ivphl -
Vo--.
L VT 2
{6)
Let the velocity spread in the beam be t:.v. A definite
Langmuir oscillation will be excited by a beam with such
a spread if the phase velocity of the oscillation lies in
the range Vo- t:.v < Vph < vo. As this oscillation propagates into the region of lower values of the density its
phase velocity will finally leave the velocity range indicated above, and the oscillation will stop growing. This
will occur over a distance
l1x-L~ v~
Vo
Vo 2
(7}
and during a time
lix
L t:J.v
-r----Vg
Vo Vo
(in order to obtain the estimate (7) one should substitute
into formula (6) t:.v in place of t:.vph· From this it can
be seen that the product y T which determines the effectiveness of the growth of oscillations is equal to
nb Vo oopL
-n~v;--
and decreases as t:. v is increased. It is clear that relaxation ceases for such values of t:.v when this product
becomes of order A i.e., when
(8)
QUASILINEAR RELAXATION OF AN ELECTRON BEAM
A rough estimate of the size of the domain of relaxation
l can be obtained by substituting this value of .:l v into
formula (7):
In this case, just as in a homogeneous plasma, relaxation does not lead to the appearance of electrons with
velocities v > vo since as the oscillations propagate in
the direction of lower values of the density their phase
velocity can only decrease. However, there also exists
an essentially new circumstance: in an inhomogeneous
plasma relaxation proceeds much less effectively than
in a homogeneous plasma. Indeed, even for nb ~ nb
0
when the relaxation in a homogeneous plasma already
leads to the formation of a plateau, in an inhomogeneous
plasma only a small velocity spread appears: .:lv/v 0
"' T/E « 1. A plateau is formed when nb exceeds the
new critical value:
As nb increases further the form of the distribution
function at the point where the beam leaves the plasma
is not altered.
We note that for nb » nb 1 inhomogeneity exerts no
influence on the relaxation process since in this case
the length of the region of relaxation which is evaluated
according to the homogeneous plasma modell ,:.,~ Lnbjnb
« LT/ & is so small that the phase velocity of the oscillations does not have time to be altered significantly
over this length (cf. (6)).
From the estimates for l given above it follows that
both for nb < nb 1 and for nb > nb 1 the dimension of the
relaxation region (i.e., the dimension over which the
distribution function attains its final state) is small
compared to L. Oscillations generated in this region
propagate freely through the plasma until they are
damped either due to collisions or due to the Cerenkov
absorption by plasma electrons.
2. THE PLASMA DENSITY INCREASES MONOTONICALLY IN THE DIRECTION OF THE INJECTION OF
THE BEAM
In this case the phase velocity of the Langmuir oscillations generated by the beam increases as they
propagate in the direction x > 0. The change in the
phase velocity over a distance .:lx can be determined as
before by formula (6). In accordance with this the estimate (8) for the final spread of the velocities in the
beam remains valid (for nb « nb)· Only the meaning of
the quantity .:lv is altered: now when the beam leaves
the plasma there appear in it not only decelerated
(v < vo), but also accelerated (v > v 0 ) electrons, and
approximately in equal numbers. In other words, the
distribution function for the beam at the point x = L
differs from zero in the velocity range v 0 - .:lv/2 < v
< Vo + .:lv/2.
The mechanism for the production of accelerated
electrons manifests itself particularly clearly for
nb » nb 1 , and we consider this case in greater detail.
As has been noted in the preceding section, under the
condition~ » ~~the inhomogeneity of the plasma has
761
no influence on the establishment of a plateau in the distribution function for the electrons of the beam since
relaxation towards the plateau state (4) occurs over a
very small distance l "'Lnb/nb « LT/ iii'. In a homogeneous plasma or in a plasma with a density which decreases monotonically in the direction of the injection
of the beam relaxation ceases at this point. But in the
case of increasing density the oscillations generated by
the beam in the region x < l and having initially phase
velocities Vph < Vo in propagating into the interior of the
plasma increase their phase velocity and lead to an acceleration of the electrons.
The phase velocity of the Langmuir oscillations
changes by an amount of the order of magnitude of itself
over a distances ~ LvT/v~ (cf., (6)). It is just over this
distance that the effect of acceleration begins to play
a significant role. On the other hand, for nb » fib 1 we
have the inequality s >> l. Therefore, the whole relaxation process can be divided into two stages: at first,
over a distance l, the establishment of a plateau occurs
in the velocity range 0 < v < vo, and the distribution
function acquires the form (4); then over a distance s,
the phase velocity of the oscillations increases and acceleration of electrons takes place.
In order to investigate the relaxation process quantitatively we utilize the following system of quasilinear
equations:
v!j_ = _!__D!J_
{)x
D=4n2 e2 W/m2 v,
(!)Po
=
{)v
L*=
Wp
(9)
{)v '
ldlnwp/dxi~'~L,
(0), no= n (0),
=
where f f(v, x) is the distribution function for the
electrons of the beam, normalized to the total number
of particles in the beam, while W = W(v, x) is the spectral density of the energy of the oscillations 4 >. In writing Eq. (9) we have utilized the fact that the size of the
domain of relaxation is small compared to the scale of
the inhomogeneity.
As has been pointed out above, the first stage of the
relaxation can be described in the approximation of a
homogeneous plasma. In other words, in considering
this stage one can omit the first term on the left hand
side of (10). The resultant system of equations has a
quasilinear integral:
Wp 0
{)
W
vf-3-vre"--= nb6(v- vo).
m
{)v v4
(11)
We have taken into account the fact that at the left-hand
boundary of the plasma the distribution function for the
electrons has the form f = nbo (v- v 0 ) and that the noise
level at the end of the first stage of relaxation is significantly higher than the thermal noise level. Substituting into Eq. (11) the function (4) for the function f(v) we
obtain the spectrum of oscillations at the end of the first
phase of relaxation:
4 >The quantity W(v,x) wpv- 2 dvdx represents the energy of oscillations contained within the range of phase velocities (v, v + dv) in a layer
of plasma of thickness dx.
B. N. BREIZMAN and D. D. RUYTOV
762
W _ W ( )_
-tV-
inbm v6
·{ --2.
3 vo
Colp, Vx
0,
v<vo
v>vo'
(12)
The further evolution of the functions f and W is described by the system (9) and (10) with the boundary conditions
for
for
v< vo
v>vo'
In order to solve (17) it is convenient to go over from
the variable v to the variable
X )-'I•,
2
v2
(
V=v 1+'3£• v~
which represents the phase velocity of a definite
Langmuir oscillation at the point x = 0. In this case (18)
assumes the form
(13)
aw
fJx
(14)
=
~ve(V x)~-1-.
3Colp,v~
(19)
d:& u2 (x)
'
where
We specify the boundary conditions at the point x = 0,
and not at the point x ~ l, since l is much smaller than
s-the characteristic scale for the second stage of the
relaxation.
We make estimates of the left hand side and the right
hand side of (9):
iJI
V 8. ¥ •
I
iJ
iJI
I
-D--D-fJv fJv
v02
Vo- ,
11
nb val
W(V,:c)= _mnbVO [
3vx 2Colp,Vo
nv~
»
0
,
v < u(x)
v > u(x)
(15)
The normalization of the distribution function is obtained from the condition of conservation of the flux of
electrons.
Since at the point x = 0 the maximum phase velocity
of the oscillations is equal to v 0 , then we have for u(x)
2 x v0•
u(x)=!lo( 1- 3 L• Vx 2
)-'I•
•
(16)
We shall see below that this formula breaks down for
3 .v x(
1)
2
x>xo=-L- 1--=- ,
2
vo2
l'3
(16')
so that the problem of the divergence of u(x) at the point
x = 3L*~T/2~ does not arise.
Since the coefficient quasilinear diffusion is great
albeit not infinite, the distribution function in the domain 0 < v < u(x) will have a nonvanishing derivative
with respect to v, i.e., it will deviate somewhat from the
plateau. The value of Bf/av can be obtained by substituting (15) into the left hand side of (9). As a result we obtain that
D!!_= 4n2e2 W!L_= _ 2nbvov2 ~.
( 17 )
iJv
m•v
fJv
u3
d:&
We see that Bf/av < 0, i.e., the oscillations are damped
as x increases. Determining from (17) the quantity
WBf/av and substituting it into the right hand side of
(10), we obtain the equation for the spectra function
W(v, x):
~ fJW + 3v2x fJW = _ 2 mnbvo• ~~. v < u(x).
(18)
L
fJv
v
fJx
(J)p,
u3
fJx
)-1/2 ,
while W is now regarded as a function of V and x. The
solution of (19) which satisfies the boundary condition
(14) can be obtained in an elementary matter:
Colp~-f.
We have taken into account Eq. (12) in accordance with
which W ~ mnbv~/w VT· From the estimates given
nb 1 the left hand
above it seems to foRow that for nb
side of (9) is much smaller than the right hand side. But
in actual fact this means that the distribution function is
near to the plateau state (i.e., jafjavl « f/v 0) over the
whole range of velocities where the diffusion coefficient
D(v, x) differs from zero (in other words, in the domain
0 < v < u(x), where by u(x) we have denoted the maximum phase velocity of oscillations at the point x):
l(v,x) ={2nbvo/u2 (x),
2 :c yz
v(V,:c)= V ( 1- 3 L' v~
=
mnbVO [
3vx'Colp,Vo
1+~f
d:&'vfl(V,x')~-1-]
VO
dx' u (x')
2
0
1+ vo• Sd:!fl 1- ~ x' ..!::)-a
~-1-J.
2
\
0
3 £• Vx
dx' u2 (x')
(20)
Under the condition (16) it follows from (20) that
W(V,x)=
2 1 -~..:__!::)-2].
mnbV6 [ 1 +.!!._- vo (
3vx2 Colp,Vo
2¥2
2¥2
3 L• vx2
It can be easily seen that the energy of those Langmuir
oscillations which at the point x = 0 had the phase veloc-
ity V = Vo,
mnbvo5 [ f - 1(
W(V,x)iv=v,=--2VxColp,
3
2 x
1----.3 L vx"
Vo2 ) ] -2
,
decreases, and at x = x 0 (cf., (16')) vanishes. With increasing penetration into the interior of the plasma
there is a progressive damping to zero of oscillations
with smaller and smaller values of the initial phase
velocity V, and at each point x > Xo only those oscillations are present the initial phase velocity of which
satisfies the inequality V < V(x), where the function V(x)
is determined by the equation
1 + vo• s" d:&'( 1- .3_~ V2(x)
3 L' Vx 2
0
)-3~-1= 0.
d:&' u•(x')
(21)
(This equation is obtained from (20)).
The position of the boundary of the plateau u = u(x) in
the region x > xo is expressed in terms of V(x):
2 x V2(x)
u(x)= V(x).( 1 - - - - 3L'vx2
)-'I•
•
(22)
Formula (16) is now, naturally, inapplicable: it could be
used only as long as the energy of the oscillations with
the initial phase velocity V = v 0 was different from zero.
From (21) and (22) we can obtain the equation for the
function u(x) which in terms of the dimensionless variables y = ~/u2 (x), ~ = 2xvV3L*vT has the form
•
i+£;+y(6)P
~
!IS'
[s-s'+Y(s)P
dy(6')
d6' =0.
(23)
We are unable to obtain an exact solution of this integral
equation, but the asymptotic value of u(x) for x - co
!lao=
limu(x),
........
QUASILINEAR RELAXATION OF AN ELECTRON BEAM
can be found very simply, even without making use of
(23). In order to do this we must utilize the fact that the
sum of the energy fluxes for the particles and the oscillations does not depend upon x and is equal to
mllbvV2. The end of the relaxation process corresponds
to a complete damping of the oscillations and the vanishing of their energy flux. On the other hand, the energy
flux for the particles is given by
763
trons, i.e., the distribution function for the electron
beam when it leaves the plasma turns out to be different
from zero in the velocity interval vo- ~v/2 ~ v ~ Vo
+ ~v/2.
Relaxation occurs within a region of width
(26)
(cf., (15)). Therefore we can write that
near the point where the plasma density has a maximum.
Formulas (25) and (26) are valid for such values of
nb when the spread ~v determined by means of the first
of these two formulas is small compared to vo, i.e., for
i.e., Uoo = vo¥2. Naturally, this result can also be obtained from (23) but by means of a lengthier argument.
It should be noted here that in actual fact the decrease in the energy of the oscillations finally leads to
the fact that the left hand side and the right hand side of
(9), formerly estimated as vof/s and Df/v~ become of
the same order of magnitude and the ''plateau approximation'' breaks down. This occurs for
As the density increases further the velocity spread increases, and the domain of relaxation gradually ''slips''
towards the left hand boundary of the plasma. Finally,
when nb becomes of the order of nb 1 relaxation is completed in a region of width of. order LT/E near the
origin. The form of the distribution function in this case
is determined by the relation (24).
~
u(x)
Sv3f(v,x)dv= mnbv~2(x)
0
4. DISCUSSION OF RESULTS
The energy flux density corresponding to this value of
W which in order of magnitude is equal to
mnbv~vo/nbLwp is small compared to the energy flux
density for the particles. Thus, the "plateau approximation'' breaks down when the energy of oscillations
becomes so small that the oscillations in practice cease
to exert an influence on the distribution function for the
electrons. Consequently, one can use the "plateau approximation" to the very end of the second stage of the
relaxation.
Summarizing the above one can assert that for
llt » nb 1 relaxation proceeds in two stages: at first the
beam excites oscillations, transfers its energy to them
and at the same time relaxes towards the state (4); then
the oscillations are again absorbed by the beam and it
arrives at the state
t={ nb/vo,
·
0
v
,
< vol'~.
> v 0l'2
v
(24)
3. THE DENSITY HAS A MAXIMUM WITHIN THE
PLASMA INTERVAL
In this case for low densities of the beam relaxation
occurs near the position of the maximum since here the
plasma density varies most slowly. In exactly the same
way as it was done above one can show that for sufficiently small values of nb the spread of the velocities is
determined by the formula
(25)
As should have been expected, the spread of the velocities increases with increasing nb more rapidly than in a
plasma with a monotonically varying density, but slower
than in a plasma with a homogeneous density. Since to
the left of the maximum the plasma density is an
increasing function of x relaxation leads to the appearance both of decelerated as well as accelerated elec-
The estimates obtained in the present paper are
qualitatively illustrated in Fig. 2. From the figure it
can be seen that the inhomogeneity of the plasma leads
to two significant effects. Firstly, when there exist in
the plasma regions with a positive derivative of the density relaxation gives rise to the appearance of accelerated electrons. Secondly, the quasilinear relaxation in
an inhomogeneous plasma proceeds at a considerably
slower rate than in a homogeneous plasma (fib 1 » fib 2
» fib 0). The last assertion refers, however, only to the
case when "trapped" oscillations are absent.
The role played by the scattering of the Langmuir
oscillations by the electrons of the plasma is not a significant one in the problem under consideration. Indeed,
the oscillations are scattered only by electrons with
very low velocities: lvl :::s v;,/vo ~ vT(T/E) 112 • In the
case when the spectrum of oscillations is one-dimensional the effect of scattering leads to the establishment
of a plateau in the distribution function for the electrons
of the plasma (in the domain of the velocities lv I
:S vT(T/ .W) 112 ). This r%quires a very small energy (in a
unit volume nT(T/ .§') 5 2 ). Therefore, after a certain time
after the beam is switched on a plateau is established in
the distribution function of the electrons of the plasma,
and after this scattering stops completely (if we do not
take pair collisions into account).
FIG. 2. Dependence of the velocity spread in the beam when it leaves
the plasma on the beam density: I the density decreases monotonically,
2 - the density increases monotonically, 3 - the density has a maximum
in the central portion of the plasma
interval. The dotted line represents
the curve corresponding to a homogeneous plasma.
764
B. N. BREIZMAN and D. D. RUYTOV
The polarization electric field E ~ T/eL which exists
in an inhomogeneous plasma is also not significant, since
a change in the phase velocity of the oscillations due to
the inhomogeneity is much greater than the change of the
velocity of the electrons of the beam in this field.
1 Ya. B. Fal:'nberg and v. D. Shapiro, Zh. Eksp. Tear.
Fiz. 47, 1389 (1964) [Sov. Phys.-JETP 20, 937 (1965)].
V. N. Tsytovich and v. D. Shapiro, Zh. Tekh. Fiz.
35, 1925 (1965) [Sov. Phys.-Tech. Phys. 10, 1485 (1966)].
3 D. D. Ryutov, Zh. Eksp. Tear. Fiz. 56, 1537 (1969)
[sic!].
4 A. P. Kropotkin, Zh. Eksp. Tear. Fiz. 53, 1765
(1967) [Sov. Phys.-JETP 26, 1011 (1968)].
2
Translated by G. Volkoff
163